Altitude estimation using a probability density function

ABSTRACT

Methods, program products, and systems of location estimation using a probability density function are disclosed. In general, in one aspect, a server can estimate an effective altitude of a wireless access gateway using harvested data. The server can harvest location data from multiple mobile devices. The harvested data can include a location of each mobile device and an identifier of a wireless access gateway that is located within a communication range of the mobile device. The server can calculate an effective altitude of the wireless access gateway using a probability density function of the harvested data. The probability density function can be a sufficient statistic of the received set of location coordinates for calculating an effective altitude of the wireless access gateway. The server can send the effective altitude of the wireless access gateway to other mobile devices for estimating altitudes of the other mobile devices.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority to U.S.application Ser. No. 13/784,664, entitled “Altitude Estimation Using aProbability Density Function,” filed Mar. 4, 2013, which claims priorityto U.S. application Ser. No. 13/152,606, entitled “Altitude EstimationUsing a Probability Density Function,” filed Jun. 3, 2011, the entirecontents of each of which are incorporated herein by reference.

TECHNICAL FIELD

This disclosure relates generally to geographic location determination.

BACKGROUND

A wireless communications network can employ various technologies formobile devices to communicate wirelessly. The wireless communicationsnetwork can include one or more wireless access gateways for connectinga mobile device to another mobile device or to a wired network. Thewireless access gateways can include, for example, cell towers orwireless access points (APs) of a wireless local area network (WLAN) ora metropolitan area network (MAN). Each of the wireless access gatewayscan serve mobile devices located in a geographic area (e.g., a cell of acellular network).

A mobile device can include one or more location-based applications thatare configured to perform location-specific tasks. A mobile deviceequipped with global positioning system (e.g., GPS) functions can use alocation determined by the global positioning system as an input to alocation-based application. A mobile device not equipped with globalpositioning system functions, or a mobile device located in an areawhere GPS signals are weak (e.g., inside buildings), can use alternativeways to determine a location. For example, if the location of a wirelessaccess gateway is known, and a mobile device is connected to thewireless access gateway, the mobile device can estimate a currentlocation using a location of the connected wireless access gateway basedon a signal strength.

SUMMARY

Methods, program products, and systems of location estimation using aprobability density function are disclosed. In general, in one aspect, aserver can estimate an effective altitude of a wireless access gatewayusing harvested data. The server can harvest location data from multiplemobile devices. The harvested data can include a location of each mobiledevice and an identifier of a wireless access gateway that is locatedwithin a communication range of the mobile device. The server cancalculate an effective altitude of the wireless access gateway using aprobability density function of the harvested data. The probabilitydensity function can be a sufficient statistic of the received set oflocation coordinates for calculating an effective altitude of thewireless access gateway. The server can send the effective altitude ofthe wireless access gateway to other mobile devices for estimatingaltitudes of the other mobile devices.

In some implementations, a server can harvest location data frommultiple mobile devices. The harvested data can include a location of amobile device and an identifier of a wireless access gateway that islocated within a communication range of the mobile device. The locationof the mobile device can include an altitude coordinate of the mobiledevice. Based on the harvested data, the server can determine aneffective altitude of the wireless access gateway using statisticalanalysis. The server can send the effective altitude of the wirelessaccess gateway to other mobile devices for estimating altitudes of theother mobile devices.

The techniques of estimating location using a probability densityfunction can be implemented to achieve the following advantages. Thelocations of wireless access gateways can be used as a supplement to GPSlocation estimation. A server can send a collection of locations ofwireless access gateways to a mobile device. When the mobile device isnot equipped with GPS functions, the mobile device can use the locationsof wireless access gateways to determine a current location. For amobile device equipped with GPS functions, the location estimation canbe used to supplement the GPS functions when the GPS signals are weak.In addition, GPS signal acquisition time can be improved over aconventional mobile device. The mobile device can estimate a locationusing the locations of wireless access gateways to assist in determiningto which GPS satellite to connect, thereby reducing the number ofchannels to scan.

Location estimation of a wireless access gateway can be more accuratecompared to conventional location determinations. The more accuratelocation estimate, in turn, can be a better basis for estimating acurrent location of a mobile device. Location estimation of a wirelessaccess gateway using a probability density function does not requiremultiple iterations in the calculation. Accordingly, location estimationof a wireless access gateway using a probability density function can beperformed faster than a conventional location determination. Theprobability density function can be a sufficient statistic of harvesteddata with respect to calculating a location. Accordingly, thecalculation can be performed using approximately the same amount ofmemory regardless of the amount of data processed.

The estimated location of a wireless access gateway can include analtitude. Thus, a mobile device can determine a current location in athree-dimensional space. The three-dimensional location information canbe useful, for example, in determining on which floor of a high-risebuilding the mobile device is located. The altitude can supplementlatitude and longitude coordinates. Accordingly, for example, analtitude profile can be created for a street or a park.

The details of one or more implementations of estimating location usinga probability density function are set forth in the accompanyingdrawings and the description below. Other features, aspects, andadvantages of estimating location using a probability density functionwill become apparent from the description, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram providing an overview of exemplary techniques oflocation estimation using a probability density function.

FIG. 2 is a diagram providing an overview of exemplary techniques oflocation estimation using a probability density function in athree-dimensional space.

FIGS. 3A and 3B are illustrations of exemplary operations of applying aprobability density function to exclude outliers in harvested data.

FIG. 4A is a top plan view of an exemplary three-dimensional histogramplot used in location estimation.

FIG. 4B is an exemplary histogram used in location estimation.

FIG. 5 is a diagram illustrating exemplary techniques of detectingmoving wireless access gateways.

FIG. 6 is flowchart illustrating exemplary operations of data harvestingand location estimation.

FIG. 7 is a block diagram illustrating various units of an exemplarysystem configured to perform location estimation using a probabilitydensity function.

FIGS. 8A-8C are flowcharts illustrating exemplary operations of locationestimation using a probability density function.

FIG. 9 is an exemplary user interface displaying an estimated location.

FIG. 10 is a block diagram illustrating an exemplary device architectureof a mobile device implementing the features and operations of locationestimation.

FIG. 11 is a block diagram of an exemplary system architecture forimplementing the features and operations of location estimation.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION Overview of Location Estimation Using a ProbabilityDensity Function

FIG. 1 is a diagram providing an overview of exemplary techniques oflocation estimation using a probability density function. A systemperforming location estimation can apply a probability density functionon data of location points distributed on geographic grid 100 toestimate an effective location of wireless access gateway 102.

An effective location of wireless access gateway 102 is a calculatedlocation of wireless access gateway 102 that can be used to calculate alocation of mobile device 104 being located within a communication rangeof wireless access gateway 102. The effective location can indicate alikely location of mobile device 104. The effective location can includelatitude, longitude, and altitude coordinates. The coordinates can beassociated with an uncertainty value, which can indicate an accuracy ofthe coordinates. The effective location can, but often does not,coincide with a physical location of wireless access gateway 102.

The system can harvest data from multiple location-aware devices 106.Each of the location-aware devices 106 can be configured to transmit acurrent location to the system anonymously. The current location caninclude a detected latitude, longitude, and altitude of thelocation-aware devices 106. The location can be associated with anidentifier of wireless access gateway 102. The identifier can include,for example, a cell identifier of wireless access gateway 102 whenwireless access gateway 102 is a cell tower, or a media access control(MAC) address when wireless access gateway 102 is a wireless accesspoint or a Bluetooth™ device. The location can be associated withadditional information relating to communication between a mobile deviceand wireless access gateway 102. The additional information can include,for example, a received signal strength indication (RSSI), bit errorrate information, or both. A data point in the harvested data caninclude the location, the identifier, and the additional information. InFIG. 1, each triangle indicates a harvested data point.

The system can use grid 100 to identify geographic regions in whichreceived locations of location-aware devices 106 are concentrated. Grid100 can be a geographic area associated with wireless access gateway 102that includes multiple tiles of geographic regions. Each tile cancorrespond to a bin into which the harvested data points can be put.Each bin is a unit in grid 100 for which a probability distribution canbe calculated. Grid 100 can include multiple bins. The system cangenerate a histogram representing a distribution of the locations in theharvested data based on the bins of grid 100. The system can select oneor more bins (e.g., bins 108 and 110) based on a probability densityfunction. The probability density function can include a sufficientstatistic of the received set of location coordinates for calculating aneffective location of wireless access gateway 102. The sufficientstatistic can include a representation of the harvested data thatretains properties of the harvested data. The sufficient statistic caninclude a likelihood technique that allows the system to model how wellthe system performs on summarizing the location coordinates in theharvested data for calculating the location of wireless access gateway102. The system can use the sufficient statistic to create a parameterthat summarizes the characteristics of the harvested data.

The system can exclude one or more bins (e.g., bin 112) that includeslocations considered outliers by the system. An outlier can be animprobable measurement unrepresentative of the harvested data. Thesystem can identify outlier 114 by identifying a location that isstatistically distant from other locations in the harvested data. When abin is excluded, the system can ignore the data points in the bin whencalculating an effective location of wireless access gateway 102.

The system can determine an effective location of wireless accessgateway 102 based on sets of locations in the selected bins 108 and 110.The system can send the effective location and effective locations ofother wireless access gateways to mobile device 104 for determining alocation of mobile device 104.

FIG. 2 is a diagram providing an overview of exemplary techniques oflocation estimation using a probability density function in athree-dimensional space. A system can determine an effective altitude ofwireless access gateway 202 using location data harvested from one ormore mobile devices 204. The effective altitude of wireless accessgateway 202 is a calculated altitude of wireless access gateway 202 thatcan be used to calculate an altitude of mobile device 208 that islocated within communication range of wireless access gateway 202. Theeffective altitude can indicate a likely altitude where mobile device208 is located. The effective altitude can be, but often is not, anactual altitude of wireless access gateway 202.

The system can create virtual layers 210, 212, 214, and 216. Eachvirtual layer can correspond to an altitude segment along a Z (altitude)axis in a three-dimensional space. Each altitude segment can have aspecified height (e.g., 10 meters). The system can generate a histogramrepresenting a distribution of the locations in the harvested data basedon virtual layers 210, 212, 214, and 216. The system can select one ormore layers (e.g., layers 210 and 216) based on a probability densityfunction. The probability density function can include a sufficientstatistic of the received set of location coordinates for calculating aneffective altitude of wireless access gateway 202.

The system can exclude one or more layers (e.g., layer 212) thatincludes one or more outliers. The system can identify outlier 218 byidentifying an altitude that is statistically distant from otheraltitudes in the harvested data.

The system can determine an effective altitude of wireless accessgateway 202 based on sets of altitudes in the selected layers 210 and216. The system can send the effective altitude and effective altitudesof other wireless access gateways to mobile device 208 for determiningan altitude of mobile device 208.

In some implementations, the system can determine an effective locationof wireless access gateway 202 in a three-dimensional space by usinglatitude, longitude, and altitude data. The system can create multipleblocks in the three-dimensional space. One of these blocks is exemplaryblock 220. Block 220 can be defined using one or more sets of latitude,longitude, and altitude coordinates that indicates a length, width, andheight. The system can generate a histogram representing a distributionof three-dimensional locations in the harvested data based on theblocks. The system can calculate a probability distribution of harvestdata points for each block. The system can select one or more blocks(e.g., blocks 222, 224, and 226) based on a probability densityfunction. The probability density function can include a sufficientstatistic of the received set of location coordinates for calculating aneffective location of wireless access gateway 202 in thethree-dimensional space. The system can calculate the effective locationof wireless access gateway 202 in the three-dimensional space by usingoperations of selection and exclusion in a similar manner as describedabove with respect to the two-dimensional and altitude calculations.

Exemplary Probability Density Function Used in Location Estimation

FIGS. 3A and 3B are illustrations exemplary operations of applying aprobability function to exclude outliers in harvested data. FIG. 3Aillustrates a conventional way of determining an effective location of awireless access gateway physically located at location “0.” Theconventional way of determining the location can include, for example,determining the location based on signal strength and triangulation. TheX axis in FIG. 3A can correspond to distance from the location. The Yaxis in FIG. 3A can correspond to a number of data sampled from variousmobile devices. A point (x, y) in FIG. 3A can indicate that based ondata from y mobile devices, the location of the wireless access gatewayis approximately x units from the y mobile devices.

The system utilizing the conventional technologies can determine aunimodal probability distribution 302 for calculating a location of thewireless access gateway. If the actual data distribution is notunimodal, the conventional system can produce suboptimal calculations.For example, information on data 304, 306, 308, and 310 indicatingconcentration far away from the average can be lost in the calculations.

FIG. 3B is a diagram illustrating calculations performed in estimating alocation using a probability density function in one dimension. The Xaxis in FIG. 3B can correspond to distance from the location of awireless access gateway physically located at location “0.” The Y axisin FIG. 3B can correspond to a probability distribution f(x) indicatingthe probability that a location coordinate in harvested data is atdistance x to the location. The probability distribution f(x) can havethe following property:

∫_(−∞) ^(∞) f(x)dx=1  (1)

The probability distribution can be multi-modal. For example, f(x) canhave local maxima 322 and 324, which will be referred to as modes off(x).

The system can determine a measurement for selecting one or more regions(e.g., regions [a, b] and [c, d]) such that an expected value in theselected region satisfies an outlier threshold. For example, the systemcan determine the measurement k using the following formula:

∫_(a) ^(b) p(x)dx+∫ _(c) ^(d) p(x)dx=1−OutlierThreshold  (2)

where

a,b,c,d=f ⁻¹(k)  (3)

In (2) and (3), a, b, c, d can define regions. P(x) can indicate alikelihood, according to harvested data, that a location coordinate islocated at distance x from an effective location. The OutlierThresholdis a threshold value below which a location coordinate in harvest datais regarded an improbable measurement and not representative of theharvested data. In some implementations, the system can solve k usingNewton's Method. In some implementations, the system can sort harvesteddata and perform the integration until the OutlierThreshold issatisfied.

The calculations are shown in a one-dimensional example. In someimplementations, the regions and corresponding calculations cancorrespond to a two-dimensional or three-dimensional space. For example,in some implementations, the regions can correspond to theone-dimensional altitude segments (as described in reference to FIG. 2),two-dimensional tiles (as described in reference to FIG. 1), orthree-dimensional blocks (as described in reference to FIG. 2).Accordingly, calculations can be multi-variable calculations. Eachaltitude segment, tile, and block can be associated with a bin.

In a two-dimensional space, the system can determine a k-th moment ofthe probability distribution based on the following formulae:

E[X ^(k)]=∫_(−∞) ^(∞)∫_(−∞) ^(∞) x ^(k) f(x,y)dydx

E[Y ^(k)]=∫_(−∞) ^(∞)∫_(−∞) ^(∞) y ^(k) f(x,y)dxdy  (4)

The system can determine expected effective location based on thefollowing formulae:

E[X]=∫ _(−∞) ^(∞)∫_(−∞) ^(∞) xf(x,y)dydx

E[Y]=∫ _(−∞) ^(∞)∫_(−∞) ^(∞) yf(x,y)dxdy  (5)

Accordingly, the system can determine the standard deviation of theeffective location using the following formulae:

E[X ² ]−E[X] ²=√{square root over (∫_(−∞) ^(∞)∫_(−∞) ^(∞) x ²f(x,y)dydx−(∫_(−∞) ^(∞)∫_(−∞) ^(∞) xf(x,y)dydx)²)}{square root over(∫_(−∞) ^(∞)∫_(−∞) ^(∞) x ² f(x,y)dydx−(∫_(−∞) ^(∞)∫_(−∞) ^(∞)xf(x,y)dydx)²)}

E[Y ² ]−E[Y] ²=√{square root over (∫_(−∞) ^(∞)∫_(−∞) ^(∞) y ²f(x,y)dxdy−(∫_(−∞) ^(∞)∫_(−∞) ^(∞) y ² f(x,y)dxdy)²)}{square root over(∫_(−∞) ^(∞)∫_(−∞) ^(∞) y ² f(x,y)dxdy−(∫_(−∞) ^(∞)∫_(−∞) ^(∞) y ²f(x,y)dxdy)²)}  (6)

FIG. 4A is a top plan view of an exemplary three-dimensional histogramplot 400 used in location estimation (hereafter referred to as“histogram 400”). Histogram 400 is implemented in a two-dimensionalspace defined by a latitude and a longitude. Other dimensions can beimplemented similar manner. Histogram 400 can be associated with awireless access gateway.

Histogram 400 can be defined using a minimum latitude, a minimumlongitude, a maximum latitude, and a maximum longitude. Size ofhistogram 400 can be determined based on technology used by the wirelessaccess gateway. For example, a histogram corresponding to a cell towercan be larger than one that corresponds to a wireless access point interms of differences between the latitudes and between the longitudes.The size of memory used in storing a larger histogram and the size ofmemory used in storing a smaller histogram can be the same.

Histogram 400 can correspond to a data structure that includescomponents as listed in Table 1 below.

TABLE 1 Histogram Data Structure DATA DESCRIPTION Device ID Anidentifier of the wireless access gateway Dimension Latitude/longitudecoordinates Width/height Counts of number of bins in longitude/latitudedimensions respectively Minimum/maximum Minimum and maximum time ofmovement. TOM Will be described in further detail below in reference toFIG. 5 Number of data Number of harvested data points in the pointshistogram Bins A list or array of bins in the histogram Minimum/maximumMinimum and maximum latitude, longitude, coordinates and altitude

Histogram 400 can include multiple bins (e.g., bins 402, 404, and 406).Some of the bins (e.g., bins 402 and 406, as represented by shaded boxesin FIG. 4) can be bins selected according to the operations as describedabove in reference to FIG. 3B. Each of the bins can be associated with adata point count (e.g., values D1 through D16 as shown in FIG. 4A). Eachof the bins can correspond to a data structure that includes componentsas listed in Table 2 below.

TABLE 2 Bin Data Structure DATA DESCRIPTION Dimension Latitude/longitudecoordinates Data points A count of number of data points in the binSignal Quality Minimum/maximum/average value of various measurements ofsignal quality of the data points (e.g., RSSI, round trip time, or biterror rate) Minimum/maximum Minimum and maximum time of movement. TOMWill be described in further detail below in reference to FIG. 5

The system can extract one or more wireless access gateway identifiersfrom harvested data, generate histogram 400 by creating the datastructures for histogram 400 and the bins in histogram 400. The systemcan populate the data structures using the harvested data, and performcalculations based on the populated data structures. The data structuresdo not depend on the number of data points harvested. Accordingly,subsequent calculations using the a probability density function neednot increase in complexity and processing time when more data points areharvested.

FIG. 4B is an exemplary histogram 440 used in location estimation.Histogram 440 can correspond to a sufficient statistic of harvested datafor calculating an effective location based on harvested data points.The sufficient statistic is shown in reference to grid 444. Histogram440 can be determined using one or more computers.

Filtering Harvested Data

FIG. 5 is a diagram illustrating exemplary techniques of detectingmoving wireless access gateways. A wireless access gateway canphysically move. For example, a wireless access point can be taken fromhome to work in the morning and from work to home in the evening. A celltower can change a corresponding cell identifier to one that originallycorresponds to another cell tower a long distance away. Identifyingmoving wireless access gateways can reduce errors in locationcalculation.

A system can identify movement of a wireless access gateway based on adistance comparison. Map 500 can include multiple grids 502 thatcorrespond to various wireless access gateways. Each grid can correspondto a wireless access gateway, and include multiple bins. Grid 504 a cancorrespond to wireless access gateway 506. Grid 504 a can have a datastructure that includes minimum and maximum latitudes, longitudes, andaltitudes. The system can determine that wireless access gateway 506 hasmoved when a distance span between two corresponding values in the gridsatisfies a threshold. In some implementations, the system can determinea movement based on altitude when the following condition is satisfied:

MaxAlt−MinAlt>AltThreshold  (7)

where MaxAlt is a maximum altitude in harvested data in a grid, MinAltis a minimum altitude in the harvested data in the grid, andAltThreshold is a specified threshold in altitude.

In some implementations, the system can determine a movement based onlatitudes and longitudes when the following condition is satisfied:

a(MaxLat−MinLat)² +b(MaxLon−MinLon)²>LatLonThreshold²  (8)

where MaxLat is a maximum altitude in harvested data in a grid, MinLatis a minimum latitude in the harvested data in the grid, MaxLon is amaximum longitude in harvested data in a grid, MinLon is a minimumlongitude in the harvested data in the grid, and AltThreshold is aspecified threshold distance. The values a and b can be weights in thelatitudes and longitudes. The default values of a and b can be 1. Thevalues of a and b can differ as the latitude goes higher. For example,in high latitude areas, the difference between MaxLon and MinLon canhave less weight than that of the difference between MaxLat and MinLat.

In some implementations, the system can determine a movement based onlatitudes, longitudes, and altitudes when the following condition issatisfied:

a(MaxLat−MinLat)² +b(MaxLon−MinLon)²+c(MaxAlt−MinAlt)²>LatLonAltThreshold²  (9)

where MaxAlt is a maximum altitude in harvested data in a grid, MinAltis a minimum altitude in the harvested data in the grid, AltThreshold isa specified threshold in altitude, MaxLat is a maximum altitude inharvested data in a grid, MinLat is a minimum latitude in the harvesteddata in the grid, MaxLon is a maximum longitude in harvested data in agrid, MinLon is a minimum longitude in the harvested data in the grid,and AltThreshold is a specified threshold distance. The values a, b, andc can be weights in the latitudes, longitudes, and altitudes.

When the system determines that a wireless access gateway has moved, thesystem can select data points from the harvested data based on agedistinctions. The system can determine a time after which a condition(7), (8), or (9) is satisfied and designate the determined time as atime of movement (TOM). The system can select data points havingtimestamps after the last TOM for location calculation, and ignore datapoints having timestamps before the last TOM. For example, before thetime of movement, the data points for wireless access gateway 506 cancorrespond to grid 504 a. After the time of movement, the data pointsfor wireless access gateway 506 can correspond to grid 504 b.

The system can determine whether wireless access gateway 506 was movingwhen data of wireless access gateway 506 were harvested. Wireless accessgateway 506, if was moving (e.g., in a car driving by a mobile devicegathering data) and was harvested by accident, can cause locationestimation errors. Accordingly, the system can exclude wireless accessgateway 506 from location calculations if wireless access gateway 506 isa moving gateway.

The system can determine movement of wireless access gateway 506 bystoring a minimum time of movement and a maximum time of movement. Thesystem can use the minimum time between movements and a maximum time ofmovement to filter out wireless access gateway 506. If the minimum timebetween the minimum time of movements and the maximum time of movementof wireless access gateway 506 satisfies a threshold (e.g., less than athreshold), the system can designate wireless access gateway 506 as alow value wireless access gateway, and excludes wireless access gateway506 from location estimation.

FIG. 6 is flowchart illustrating exemplary operations of data harvestingand location estimation. The operations can include data harvestingoperations 600 and location estimation operations 602. Data harvestingoperations 600 can be performed continuously, for example, as a daemon.Data harvesting operations 600 can be performed upon data arrival. Asystem can parse (604) the data when the data arrive. Parsing the datacan include identifying data fields for latitude, longitude, altitude,timestamp, wireless access gateway identifier, RSSI, or otherinformation.

The system can register (606) the parsed data as harvested data.Registering the parsed data can include storing at least a portion ofthe parsed data in a data store. Registering the parsed data can includeexcluding some of the parsed data when the parsed data includes invalidinformation (e.g., an invalid wireless access gateway identifier).

The system can filter (608) the harvested data. Filtering the harvesteddata can include identifying stale data that the system will no longeruse to estimate a location and discarding the identified stale data. Thestale data can include location data corresponding to a wireless accessgateway that has moved.

The system can perform location estimation operations 602 under a schemethat is independent from the data harvesting operations 600. Forexample, the system can perform location estimation operations 602periodically (e.g., every two weeks) or upon request. The system canretrieve (610) the harvested data. The operations of retrievingharvested data can include interacting with operations of registeringthe data (operations 606). The system can estimate (612) a location of awireless access gateway using retrieved data.

Exemplary System Components

FIG. 7 is a block diagram illustrating various units of an exemplarysystem configured to perform location estimation using a probabilitydensity function. Location estimation system 700 can include dataharvesting unit 702. Data harvesting unit 702 is a component of locationestimation system 700 that is programmed to receive and process datafrom one or more mobile devices 704. Data harvesting unit 702 caninclude data parsing unit 706. Data parsing unit 706 is a component ofdata harvesting unit 702 that is configured to receive the raw data fromthe one or more mobile devices 704, parse the data fields of the rawdata, and generate structured data (e.g., name-value pairs). Furtherdetails of operations of data parsing unit 706 are described above inreference to stage 604 of FIG. 6.

Data harvesting unit 702 can include data registration unit 708. Dataregistration unit 708 is a component of data harvesting unit 702 that isconfigured to receive parsed data (e.g., name-value pairs) generated bydata parsing unit 706, and send at least a portion of the parsed data todata point data store 710 for storage. Further details of operations ofregistration unit 708 are described above in reference to stage 606 ofFIG. 6. Data point data store 710 can include a database (e.g., arelational database, an object-oriented database, or a flat file) thatis configured to store location information in association with wirelessaccess gateway identifiers.

Data harvesting unit 702 can include data filtering unit 712. Datafiltering unit 712 is a component of data harvesting unit 702 that isconfigured to identify stale data from data point data store 710, andremove the stale data from data point data store 710. Further details ofoperations of filtering unit 712 are described above in reference toFIG. 5 and stage 608 of FIG. 6.

Location estimation system 700 can include location calculation unit714. Location calculation unit 714 is a component of location estimationsystem 700 that is configured to generate one or more estimatedlocations based on data points stored in data point data store 710 usinga probability density function. Location calculation unit 714 caninclude histogram generation unit 716. Histogram generation unit 716 isa component of location calculation unit 714 that is configured togenerate a histogram (e.g., histogram 400 as described in reference toFIG. 4A) based on data points from data point data store 710. Histogramgeneration unit 716 can generate a histogram for each wireless accessgateway.

Location calculation unit 714 can include grid selection unit 718. Gridselection unit 718 is a component of location calculation unit 714 thatis configured to select one or more bins from the histogram generated byhistogram generation unit 716 using a probability density function. Theselection operations can include applying the probability function asdescribed above in reference to FIG. 3B.

Location calculation unit 714 can include location calculator 720.Location calculator 720 is a component of location calculation unit 714that is configured to calculate a location of each wireless accessgateway based on the selected bins, and to calculate an uncertainty ofthe calculated location. The calculated location can include locationcoordinates including a latitude, a longitude, and an altitude. Theuncertainty can indicate an estimated accuracy of the calculatedlocation.

Location calculator 720 can be configured to calculate a reach of eachwireless access from information associated with data points stored indata point data store 710. The reach of a wireless access gateway canindicate a maximum distance from which the wireless access gateway canbe expected to be observed by a mobile device. Location calculator 720can calculate the reach using locations in the harvested data and thecalculated location.

Location calculation unit 714 can generate output including the locationcoordinates determined by location calculator 720. The locationcoordinates can be associated with an identifier of the wireless accessgateway, an uncertainty, and a reach of the wireless access gateway.Location estimation system 700 can store the output in a location datastore 722. Location data store 722 can be a database configured to storethe location coordinates and associated information.

Location estimation system 700 can include data distribution unit 724.Data distribution 724 is a component of location estimation system 700that is configured to retrieve the location coordinates and associatedinformation stored in location data store 722 and send the locationcoordinates and associated information to one or more mobile devices726. Mobile devices 726 can be the same mobile devices as mobile device704, or separate and different mobile devices.

Exemplary Operations of Location Estimation

FIGS. 8A-8C are flowcharts illustrating exemplary operations 800 oflocation estimation using a probability density function. FIG. 8A is aflowchart illustrating exemplary operations of location estimation usinga sufficient statistic of harvested data for calculating an effectivelocation. Operations 800 of FIG. 8A can be performed by a systemincluding hardware and software components (e.g., location estimationsystem 700 as described above in reference to FIG. 7).

The system can receive (802) multiple sets of location coordinates fromone or more mobile devices. Each set of location coordinates can beassociated with a wireless access gateway. Each set of locationcoordinates can include a latitude, a longitude, and an altitude. Thealtitude can be measured in meters or feet from sea level. The wirelessaccess gateway can include a wireless device operable to connect amobile device to at least one of a personal area network, a local areanetwork, a metropolitan area network, a wide area network, or a cellularnetwork. For example, the wireless access gateway can include an AP, acell tower, or a Bluetooth™ device.

The system can map (804) the sets of location coordinates to multiplegeographic regions. In some implementations, each geographic region canbe a bin of a geographic grid comprising multiple bins. The geographicgrid can be a geographic area associated with the wireless accessgateway.

The system can select (806) one or more geographic regions from themultiple geographic regions. The selection can be based on a density ofreceived location coordinates in each of the geographic regions.Selecting the one or more geographic regions can be based on a specifiedoutlier threshold for identifying and excluding one or more outliers inthe sets of location coordinates.

The system can perform the selection operations using a probabilitydensity function. The probability density function can include asufficient statistic of the received set of location coordinates forcalculating an effective location of the wireless access gateway.Selecting the one or more geographic regions can include, determining,for each geographic region and using the probability density function,an expected value based on a relative probability that a received set oflocation coordinates is located within the geographic region. The systemcan select the one or more geographic regions when a measurement of theexpected value corresponding to the one or more geographic regionssatisfies the outlier threshold. The measurement can be a sum orweighted sum. The system can determine that the measurement satisfiesthe outlier threshold when a sum or weighted sum of the correspondingexpected values equals one minus the outlier threshold. Further detailson operations of determining that the measurement satisfies the outlierthreshold are described above in reference to FIG. 3B.

In some implementations, each set of the location coordinates isassociated with a weight, the weight indicating a degree of certainty ofthe set of location coordinates. The expected value can be determinedbased on the relative probability and the weight. The system candetermine the weight based on at least one of a received signal strengthindication (RSSI) or a bit error rate associated with each data point.Applying the weights, the system can determine a k-th moment of theprobability distribution based on the following formula:

In some implementations, the system can select one or more sets oflocation coordinates from the selected one or more geographic regionsbased on an estimated movement of the wireless access gateway.Determining the effective location of the wireless access gateway caninclude determining the effective location of the wireless accessgateway using the selected sets of location coordinates. Selecting theone or more sets of location coordinates from the selected one or moregeographic regions can include determining that at least one set oflocation coordinates is obsolete when a variation of sets of locationcoordinates exceeds a threshold. The variation of sets of locationcoordinates can exceed the threshold when the wireless access gatewayhas moved. The system can select the one or more sets of locationcoordinates by excluding the obsolete set of location coordinates.

To determine the variation, the system can utilize timestamps. Each setof location coordinates can have a timestamp corresponding to a time ofmeasurement. Selecting the one or more sets of location coordinates caninclude excluding a collection of one or more sets of locationcoordinates in a geographic region when a span of the corresponding timeof measurements of the sets in the collection satisfies a thresholdtime.

The system can determine (808) the effective location of the wirelessaccess gateway using sets of location coordinates in the selected one ormore geographic regions. The effective location can include a reach ofthe wireless access gateway and an estimated uncertainty of the wirelessaccess gateway. The system can send the effective location to one ormore mobile devices. A mobile device located within a communicationrange of the wireless access gateway can use the effective location tocalculate a current location of the mobile device.

FIG. 8B is a flowchart illustrating exemplary operations 820 of altitudeestimation based on statistics analysis. A system for determining aneffective altitude of a wireless access gateway can receive (822)multiple sets of location coordinates from one or more mobile devices.Each set of location coordinates can be associated with a wirelessaccess gateway. Each set of location coordinates can include analtitude.

The system can determine (824) an effective altitude of the wirelessaccess gateway based on a statistical analysis using the received setsof location coordinates. Further details on determining the effectivealtitude of the wireless access gateway based on a statistical analysiswill be described below in reference to FIG. 8C.

The system can provide (826) the determined effective altitude to amobile device for determining an altitude of the mobile device when themobile device is located within a communication range of the wirelessaccess gateway.

FIG. 8C is a flowchart illustrating exemplary operations 824 ofdetermine an effective altitude of the wireless access gateway based ona statistical analysis. A system can map (842) sets of locationcoordinates to multiple elevation segments.

The system can select (844) one or more elevation from the multipleelevation segments based on a density of received location coordinatesin each of the elevation segments using a probability density function.The probability density function can include a sufficient statistic ofthe received sets of location coordinates for calculating the effectivealtitude. Selecting the one or more elevation segments can includedetermining, for each elevation segment and using the probabilitydensity function, an expected value based on a relative probability thata received set of location coordinates is located within the elevationsegment. The system can select the one or more elevation segments when ameasurement of the expected probability value corresponding to the oneor more elevation segments satisfies an outlier threshold. The systemcan determine that the measurement satisfies the outlier threshold whena sum or weighted sum of the corresponding expected values equals oneminus the outlier threshold.

In some implementations, the system can select one or more sets oflocation coordinates from the selected one or more elevation segmentsbased on an estimated movement of the wireless access gateway.Determining the effective altitude of the wireless access gateway caninclude determining the effective altitude of the wireless accessgateway using the selected sets of location coordinates. Selecting theone or more sets of location coordinates from the selected one or moreelevation segments can include determining that at least one set oflocation coordinates is obsolete when a variation of sets of locationcoordinates exceeds a threshold. The variation of sets of locationcoordinates can exceed the threshold when the wireless access gatewayhas moved. The system can select the one or more sets of locationcoordinates by excluding the obsolete set of location coordinates.

To determine the variation, the system can utilize timestamps. Each setof location coordinates can have a timestamp corresponding to a time ofmeasurement. Selecting the one or more sets of location coordinates caninclude excluding a collection of one or more sets of locationcoordinates in a elevation segment when a span of the corresponding timeof measurements of the sets in the collection satisfies a thresholdtime.

The system can determine (846) the effective altitude of the wirelessaccess gateway using sets of location coordinates in the selected one ormore elevation segments. The system can send the effective altitude ofthe wireless access gateway to one or more mobile devices for estimatingan altitude of the mobile devices.

Exemplary User Interface

FIG. 9 is an exemplary user interface of a mobile device utilizing theestimated locations of a wireless access gateway. Mobile device 726 caninclude a touch-sensitive display device 930. Mobile device 726 candisplay map 902 of a geographic area on touch-sensitive display device930.

The search bar 904 can be used to find an address or other location onthe map. For example, a user can enter their home address in the searchbar 904, and the region containing the address would be displayed on themap 902. The bookmarks list object 906 can, for example, bring up aBookmarks list that contains addresses that are frequently visited, suchas a user's home address. The Bookmarks list can also, for example,contain special bookmarks such as the current location (e.g. the currentlocation of mobile device 726).

The search object 908 can be used to display the search bar 904 andother map related search menus. The directions object 910 can, forexample, bring up a menu interface that allows the user to enter a startand end location. The interface can then display information (e.g.,directions and travel time for a route from the start location to theend location). The map view object 912 can bring up a menu that willallow the user to select display options for the map 902. For example,the map 902 can be changed from black and white to color, the backgroundof the map can be changed, or the user can change the brightness of themap.

The current location object 914 can allow the user to see a geographicarea 916 on the map 902 indicating where mobile device 726 is currentlylocated. Geographic area 916 can correspond to an estimated geographiclocation. The estimated location can be determined based on effectivelocations of wireless access gateways that are within communicationrange of mobile device 726. A special current location bookmark can beplaced in the Bookmarks list when the current location object 914 isselected. If the special current location bookmark was previously set inthe Bookmarks list, the old bookmark information can, for example, bereplaced with the new current location information. In someimplementations, the special current location bookmark is tied to thecentroid of geographic area 916. That is, the special current locationbookmark can include the coordinates for the centroid of the geographicarea 916. The geographic area 916 can be based on location datadetermined or estimated using location instructions stored in a memorydevice of mobile device 726. The geographic area 916 can, for example,be depicted by a circle, rectangle, square, hexagon, or other enclosedregion with crosshairs, or some other distinctive element todifferentiate the geographic area 916 from the map 902.

Exemplary Mobile Device Architecture

FIG. 10 is a block diagram illustrating an exemplary device architecture1000 of a mobile device implementing the features and operations ofsending location data to a server and determining a current locationusing wireless access gateways. A mobile device can include memoryinterface 1002, one or more data processors, image processors and/orprocessors 1004, and peripherals interface 1006. Memory interface 1002,one or more processors 1004 and/or peripherals interface 1006 can beseparate components or can be integrated in one or more integratedcircuits. Processors 1004 can include one or more application processors(APs) and one or more baseband processors (BPs). The applicationprocessors and baseband processors can be integrated in one singleprocess chip. The various components in mobile device 726, for example,can be coupled by one or more communication buses or signal lines.

Sensors, devices, and subsystems can be coupled to peripherals interface1006 to facilitate multiple functionalities. For example, motion sensor1010, light sensor 1012, and proximity sensor 1014 can be coupled toperipherals interface 1006 to facilitate orientation, lighting, andproximity functions of the mobile device. Location processor 1015 (e.g.,GPS receiver) can be connected to peripherals interface 1006 to providegeopositioning. Electronic magnetometer 1016 (e.g., an integratedcircuit chip) can also be connected to peripherals interface 1006 toprovide data that can be used to determine the direction of magneticNorth. Thus, electronic magnetometer 1016 can be used as an electroniccompass. Gravimeter 1017 can include one or more devices connected toperipherals interface 1106 and configured to measure a localgravitational field of Earth.

Camera subsystem 1020 and an optical sensor 1022, e.g., a chargedcoupled device (CCD) or a complementary metal-oxide semiconductor (CMOS)optical sensor, can be utilized to facilitate camera functions, such asrecording photographs and video clips.

Communication functions can be facilitated through one or more wirelesscommunication subsystems 1024, which can include radio frequencyreceivers and transmitters and/or optical (e.g., infrared) receivers andtransmitters. The specific design and implementation of thecommunication subsystem 1024 can depend on the communication network(s)over which a mobile device is intended to operate. For example, a mobiledevice can include communication subsystems 1024 designed to operateover a CDMA system, a WiFi™ or WiMax™ network, and a Bluetooth™ network.In particular, the wireless communication subsystems 1024 can includehosting protocols such that the mobile device can be configured as abase station for other wireless devices.

Audio subsystem 1026 can be coupled to a speaker 1028 and a microphone1030 to facilitate voice-enabled functions, such as voice recognition,voice replication, digital recording, and telephony functions.

I/O subsystem 1040 can include touch screen controller 1042 and/or otherinput controller(s) 1044. Touch-screen controller 1042 can be coupled toa touch screen 1046 or pad. Touch screen 1046 and touch screencontroller 1042 can, for example, detect contact and movement or breakthereof using any of a plurality of touch sensitivity technologies,including but not limited to capacitive, resistive, infrared, andsurface acoustic wave technologies, as well as other proximity sensorarrays or other elements for determining one or more points of contactwith touch screen 1046.

Other input controller(s) 1044 can be coupled to other input/controldevices 1048, such as one or more buttons, rocker switches, thumb-wheel,infrared port, USB port, and/or a pointer device such as a stylus. Theone or more buttons (not shown) can include an up/down button for volumecontrol of speaker 1028 and/or microphone 1030.

In one implementation, a pressing of the button for a first duration maydisengage a lock of the touch screen 1046; and a pressing of the buttonfor a second duration that is longer than the first duration may turnpower to mobile device 726 on or off. The user may be able to customizea functionality of one or more of the buttons. The touch screen 1046can, for example, also be used to implement virtual or soft buttonsand/or a keyboard.

In some implementations, mobile device 726 can present recorded audioand/or video files, such as MP3, AAC, and MPEG files. In someimplementations, mobile device 726 can include the functionality of anMP3 player. Mobile device 726 may, therefore, include a pin connectorthat is compatible with the iPod. Other input/output and control devicescan also be used.

Memory interface 1002 can be coupled to memory 1050. Memory 1050 caninclude high-speed random access memory and/or non-volatile memory, suchas one or more magnetic disk storage devices, one or more opticalstorage devices, and/or flash memory (e.g., NAND, NOR). Memory 1050 canstore operating system 1052, such as Darwin, RTXC, LINUX, UNIX, OS X,WINDOWS, or an embedded operating system such as VxWorks. Operatingsystem 1052 may include instructions for handling basic system servicesand for performing hardware dependent tasks. In some implementations,operating system 1052 can include a kernel (e.g., UNIX kernel).

Memory 1050 may also store communication instructions 1054 to facilitatecommunicating with one or more additional devices, one or more computersand/or one or more servers. Memory 1050 may include graphical userinterface instructions 1056 to facilitate graphic user interfaceprocessing; sensor processing instructions 1058 to facilitatesensor-related processing and functions; phone instructions 1060 tofacilitate phone-related processes and functions; electronic messaginginstructions 1062 to facilitate electronic-messaging related processesand functions; web browsing instructions 1064 to facilitate webbrowsing-related processes and functions; media processing instructions1066 to facilitate media processing-related processes and functions;GPS/Navigation instructions 1068 to facilitate GPS andnavigation-related processes and instructions; camera instructions 1070to facilitate camera-related processes and functions; magnetometer data1072 and calibration instructions 1074 to facilitate magnetometercalibration. The memory 1050 may also store other software instructions(not shown), such as security instructions, web video instructions tofacilitate web video-related processes and functions, and/or webshopping instructions to facilitate web shopping-related processes andfunctions. In some implementations, the media processing instructions1066 are divided into audio processing instructions and video processinginstructions to facilitate audio processing-related processes andfunctions and video processing-related processes and functions,respectively. An activation record and International Mobile EquipmentIdentity (IMEI) or similar hardware identifier can also be stored inmemory 1050. Memory 1050 can include location instructions 1076.Location instructions 1076 can be a computer program product that isconfigured to cause the mobile device to anonymously send a currentlocation to a server.

Each of the above identified instructions and applications cancorrespond to a set of instructions for performing one or more functionsdescribed above. These instructions need not be implemented as separatesoftware programs, procedures, or modules. Memory 1050 can includeadditional instructions or fewer instructions. Furthermore, variousfunctions of the mobile device may be implemented in hardware and/or insoftware, including in one or more signal processing and/or applicationspecific integrated circuits.

Exemplary System Architecture

FIG. 11 is a block diagram of an exemplary system architecture forimplementing the features and operations location estimation based on aprobability density function. Other architectures are possible,including architectures with more or fewer components. In someimplementations, architecture 1100 includes one or more processors 1102(e.g., dual-core Intel® Xeon® Processors), one or more output devices1104 (e.g., LCD), one or more network interfaces 1106, one or more inputdevices 1108 (e.g., mouse, keyboard, touch-sensitive display) and one ormore computer-readable mediums 1112 (e.g., RAM, ROM, SDRAM, hard disk,optical disk, flash memory, etc.). These components can exchangecommunications and data over one or more communication channels 1110(e.g., buses), which can utilize various hardware and software forfacilitating the transfer of data and control signals betweencomponents.

The term “computer-readable medium” refers to any medium thatparticipates in providing instructions to processor 1102 for execution,including without limitation, non-volatile media (e.g., optical ormagnetic disks), volatile media (e.g., memory) and transmission media.Transmission media includes, without limitation, coaxial cables, copperwire and fiber optics.

Computer-readable medium 1112 can further include operating system 1114(e.g., Mac OS® server, Windows® NT server), network communication module1118, database interface 1120, data collection module 1130, locationcalculation module 1140, and data distribution module 1150. Databaseinterface 1120 can provide functions for various data stores for storinglocation data. Data collection module 1130 can be configured to locationcoordinates and identifiers of wireless access gateways from mobiledevices. Location estimation module 1140 can be configured to determinean effective location to be associated with each wireless accessgateway. Data distribution module 1150 can be configured to distributethe effective location and associated identifiers of wireless accessgateways to mobile devices. Operating system 1114 can be multi-user,multiprocessing, multitasking, multithreading, real time, etc. Operatingsystem 1114 performs basic tasks, including but not limited to:recognizing input from and providing output to devices 1106, 1108;keeping track and managing files and directories on computer-readablemediums 1112 (e.g., memory or a storage device); controlling peripheraldevices; and managing traffic on the one or more communication channels1110. Network communications module 1118 includes various components forestablishing and maintaining network connections (e.g., software forimplementing communication protocols, such as TCP/IP, HTTP, etc.).Database interface 1120 can include interfaces to one or more databaseson a file system. The databases can be organized under a hierarchicalfolder structure, the folders mapping to directories in the file system.Data collection module 1130 can include components for collecting datafrom multiple mobile devices.

Architecture 1100 can be implemented in a parallel processing orpeer-to-peer infrastructure or on a single device with one or moreprocessors. Software can include multiple software components or can bea single body of code.

The described features can be implemented advantageously in one or morecomputer programs that are executable on a programmable system includingat least one programmable processor coupled to receive data andinstructions from, and to transmit data and instructions to, a datastorage system, at least one input device, and at least one outputdevice. A computer program is a set of instructions that can be used,directly or indirectly, in a computer to perform a certain activity orbring about a certain result. A computer program can be written in anyform of programming language (e.g., Objective-C, Java), includingcompiled or interpreted languages, and it can be deployed in any form,including as a stand-alone program or as a module, component,subroutine, a browser-based web application, or other unit suitable foruse in a computing environment.

Suitable processors for the execution of a program of instructionsinclude, by way of example, both general and special purposemicroprocessors, and the sole processor or one of multiple processors orcores, of any kind of computer. Generally, a processor will receiveinstructions and data from a read-only memory or a random access memoryor both. The essential elements of a computer are a processor forexecuting instructions and one or more memories for storing instructionsand data. Generally, a computer will also include, or be operativelycoupled to communicate with, one or more mass storage devices forstoring data files; such devices include magnetic disks, such asinternal hard disks and removable disks; magneto-optical disks; andoptical disks. Storage devices suitable for tangibly embodying computerprogram instructions and data include all forms of non-volatile memory,including by way of example semiconductor memory devices, such as EPROM,EEPROM, and flash memory devices; magnetic disks such as internal harddisks and removable disks; magneto-optical disks; and CD-ROM and DVD-ROMdisks. The processor and the memory can be supplemented by, orincorporated in, ASICs (application-specific integrated circuits).

To provide for interaction with a user, the features can be implementedon a computer having a display device such as a CRT (cathode ray tube)or LCD (liquid crystal display) monitor for displaying information tothe user and a keyboard and a pointing device such as a mouse or atrackball by which the user can provide input to the computer.

The features can be implemented in a computer system that includes aback-end component, such as a data server, or that includes a middlewarecomponent, such as an application server or an Internet server, or thatincludes a front-end component, such as a client computer having agraphical user interface or an Internet browser, or any combination ofthem. The components of the system can be connected by any form ormedium of digital data communication such as a communication network.Examples of communication networks include, e.g., a LAN, a WAN, and thecomputers and networks forming the Internet.

The computer system can include clients and servers. A client and serverare generally remote from each other and typically interact through anetwork. The relationship of client and server arises by virtue ofcomputer programs running on the respective computers and having aclient-server relationship to each other.

A number of implementations of the invention have been described.Nevertheless, it will be understood that various modifications can bemade without departing from the spirit and scope of the invention.

What is claimed is:
 1. A method performed comprising: receiving, by oneor more computers and from a plurality of mobile devices, harvested dataabout a wireless access gateway, the harvested data including respectivelocations of the mobile devices recorded by the mobile devices when therespective mobile devices were in communication range of the wirelessaccess gateway, each location being represented using respectivelatitude and longitude coordinates; determining, by the one or morecomputers, a value of a function, the function weighting a firstdifference between latitude coordinates of two locations in theharvested data and a second difference between longitude coordinates oftwo locations in the harvested data; and in response to determining thatthe value exceeds a distance threshold, designating, in determining alocation of a mobile device using locations of one or more wirelessaccess gateways, the wireless access gateway as a moving wireless accessgateway.
 2. The method of claim 1, wherein the wireless access gatewayis a wireless access point, a cellular tower, or a Bluetooth™ device. 3.The method of claim 1, wherein: the first difference between thelatitude coordinates is given a first weight in the function, the seconddifference between longitude coordinates is given a second weight in thefunction, and higher latitudes corresponds to higher first weight overthe second weight.
 4. The method of claim 1, comprising, upondesignating the wireless access gateway as a moving wireless accessgateway: determining a time of movement of the wireless access gateway;determining a geographic grid representing a location of the wirelessaccess gateway at a time after the time of movement, the geographic gridbeing associated with a probability density function indicating arespective probability that the mobile device is located at a pluralityof bin locations in the grid; and providing a record of the geographicgrid and the probability density function to the mobile device fordetermining the location of the mobile device.
 5. The method of claim 4,wherein determining the geographic grid comprises filtering theharvested data by the time of movement, including selecting, from theharvested data, locations having timestamps after the time of movementfor determining the geographic grid.
 6. The method of claim 1,comprising, upon designating the wireless access gateway as a movingwireless access gateway: determining a first time of movement of thewireless access gateway and a second time of movement of the wirelessaccess gateway; determining a difference between the first time ofmovement and the second time of movement; and upon determining that thedifference between the first time of movement and the second time ofmovement is less than an exclusion threshold, designating the wirelessaccess gateway as a low value wireless access gateway and excluding thewireless access gateway from future location estimation.
 7. The methodof claim 1, wherein: each location is further represented using arespective altitude coordinate, and the function further weights a thirddifference between altitude coordinates between two locations in theharvested data.
 8. The method of claim 7, wherein the function squareseach of the first difference, second difference, and third differencebefore weighting.